The Stacks project

Lemma 66.23.3. Let $S$ be a scheme. Let $X$ be a decent Jacobson algebraic space over $S$. Then $X_{\text{ft-pts}} \subset |X|$ is the set of closed points.

Proof. If $x \in |X|$ is closed, then we can represent $x$ by a closed immersion $\mathop{\mathrm{Spec}}(k) \to X$, see Lemma 66.14.6. Hence $x$ is certainly a finite type point.

Conversely, let $x \in |X|$ be a finite type point. We know that $x$ can be represented by a quasi-compact monomorphism $\mathop{\mathrm{Spec}}(k) \to X$ where $k$ is a field (Definition 66.6.1). On the other hand, by definition, there exists a morphism $\mathop{\mathrm{Spec}}(k') \to X$ which is locally of finite type and represents $x$ (Morphisms, Definition 29.16.3). We obtain a factorization $\mathop{\mathrm{Spec}}(k') \to \mathop{\mathrm{Spec}}(k) \to X$. Let $U \to X$ be any étale morphism with $U$ affine and consider the morphisms

\[ \mathop{\mathrm{Spec}}(k') \times _ X U \to \mathop{\mathrm{Spec}}(k) \times _ X U \to U \]

The quasi-compact scheme $\mathop{\mathrm{Spec}}(k) \times _ X U$ is étale over $\mathop{\mathrm{Spec}}(k)$ hence is a finite disjoint union of spectra of fields (Remark 66.4.1). Moreover, the first morphism is surjective and locally of finite type (Morphisms, Lemma 29.15.8) hence surjective on finite type points (Morphisms, Lemma 29.16.6) and the composition (which is locally of finite type) sends finite type points to closed points as $U$ is Jacobson (Morphisms, Lemma 29.16.8). Thus the image of $\mathop{\mathrm{Spec}}(k) \times _ X U \to U$ is a finite set of closed points hence closed. Since this is true for every affine $U$ and étale morphism $U \to X$, we conclude that $x \in |X|$ is closed. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BA5. Beware of the difference between the letter 'O' and the digit '0'.